The problem provides the following angle measures:

• $m \angle LKQ = 3x + 11$
• $m \angle LKJ = 21x - 3$
• $m \angle QKJ = 130^\circ$

From the diagram (and typical geometric notation), we can assume that these angles form a triangle, where the sum of the angles in any triangle is $180^\circ$.

Therefore, we can write the equation:

$m \angle LKQ + m \angle LKJ + m \angle QKJ = 180^\circ$

Substituting the given values:

$(3x + 11) + (21x - 3) + 130 = 180$

Now, we can simplify and solve for $x$:

1. Combine like terms: $3x + 21x + 11 - 3 + 130 = 180$ $24x + 138 = 180$

2. Subtract 138 from both sides: $24x = 42$

3. Divide both sides by 24: $x = \frac{42}{24} = \frac{7}{4} = 1.75$

So, $x = 1.75$.

Let me know if you need more details or have any questions!

Follow-up questions:

1. What is the measure of $m \angle LKQ$ when $x = 1.75$?
2. What is the measure of $m \angle LKJ$ when $x = 1.75$?
3. How can you verify if the angle measures add up to $180^\circ$?
4. Can this method apply to any triangle with given expressions for the angles?
5. What would happen if the sum of the angles exceeded $180^\circ$?

Tip:

Always check that your final angle values add up to $180^\circ$ in any triangle problem to ensure correctness!